Divergence of separated nets with respect to displacement equivalence

Michael Dymond, Vojtěch Kaluža*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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Abstract

We introduce a hierarchy of equivalence relations on the set of separated nets of a given Euclidean space, indexed by concave increasing functions ϕ:(0, ∞)→(0, ∞). Two separated nets are called ϕ-displacement equivalent if, roughly speaking, there is a bijection between them which, for large radii R, displaces points of norm at most R by something of order at most ϕ(R). We show that the spectrum of ϕ-displacement equivalence spans from the established notion of bounded displacement equivalence, which corresponds to bounded ϕ, to the indiscrete equivalence relation, corresponding to ϕ(R)∈Ω(R), in which all separated nets are equivalent. In between the two ends of this spectrum, the notions of ϕ-displacement equivalence are shown to be pairwise distinct with respect to the asymptotic classes of ϕ(R) for R→∞. We further undertake a comparison of our notion of ϕ-displacement equivalence with previously studied relations on separated nets. Particular attention is given to the interaction of the notions of ϕ-displacement equivalence with that of bilipschitz equivalence.
Original languageEnglish
Article number15
JournalGeometriae Dedicata
Volume218
Issue number1
Early online date17 Nov 2023
DOIs
Publication statusPublished - Feb 2024

Bibliographical note

Funding:
Open access funding provided by Institute of Science and Technology (IST Austria). This work was started while both authors were employed at the University of Innsbruck and enjoyed the full support of Austrian Science Fund (FWF): P 30902-N35. It was continued when the first named author was employed at University of Leipzig and the second named author was employed at Institute of Science and Technology of Austria, where he was supported by an IST Fellowship.

Keywords

  • Density
  • ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-regularity
  • 26B35
  • 51M05
  • Displacement equivalence
  • Bilipschitz equivalence
  • 26B10
  • 51F99
  • Separated net
  • 52C99

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